Finite Group Databasebeta

← #25126312

Subgroups — #25126312

Order 11008.

808 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13063350C211yesmaximal, characteristic, complemented
2C2×#14583009C211yesmaximal, characteristic, complemented
3#19143246C211yesmaximal, characteristic, complemented
4#19136108C211yesmaximal, characteristic, complemented
5#19136043C211yesmaximal, characteristic, complemented
6#19135955C211yesmaximal, characteristic, complemented
7#19135952C211yesmaximal, characteristic, complemented
8#19135885C211yesmaximal, characteristic, complemented
9#19135884C211yesmaximal, characteristic, complemented
10#19135850C211yesmaximal, characteristic, complemented
11#19135701C211yesmaximal, characteristic
12#19134502C211yesmaximal, characteristic, complemented
13C4×#13063241C211yesmaximal, characteristic, complemented
14C4×#13063242C211yesmaximal, characteristic, complemented
15C4×AGL(1,ℤ/8)×C43C211yesmaximal, characteristic, complemented
16C2×#13063350C2211yescharacteristic
17C22×#10621882C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19#14583009—22no—
20#14583009—22no—
21#14583009—21no—
22#14582997—21no—
23C4×#10621884—21no—
24C4×#10621882—21no—
25#14582960—22no—
26#14582956—22no—
27C2×#13063260C2211yescharacteristic
28C2×#13063260C2211yescharacteristic
29#14582942C2211yescharacteristic
30C2×#13063256C2211yescharacteristic
31#14582926C2211yescharacteristic
32#14582926C2211yescharacteristic
33#14582903C2211yescharacteristic
34#14582900C2211yescharacteristic
35C2×#13063246C2211yescharacteristic
36C2×#13063245C2211yescharacteristic
37C2×#13063242C2211yescharacteristic
38C2×#13063242C2211yescharacteristic
39C2×#13063241C2211yescharacteristic
40C2×#13063241C2211yescharacteristic
41#14582866C2211yescharacteristic
42#14582866C2211yescharacteristic
43C4×#10621877C2211yescharacteristic
44C4×#10621877C2211yescharacteristic
45C2×#13063236C2211yescharacteristic
46#14582668C2211yescharacteristic
47#14582668C2211yescharacteristic
48#14582667C2211yescharacteristic
49#14582667C2211yescharacteristic
50#14582663C2211yescharacteristic
51#14582663C2211yescharacteristic
52C2×AGL(1,ℤ/8)×C43C2211yescharacteristic
53C4×#41×C43C2211yescharacteristic
54C2×C4×D4×C43C2211yescharacteristic
55C43×#442—21no—
56C43×#440—21no—
57C4×SD16×C43—21no—
58C4×D8×C43—21no—
59C43×#428C2211yescharacteristic
60C43×#418C2211yescharacteristic
61C43×#417C2211yescharacteristic
62C4×M16×C43C2211yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65#13063350—22no—
66#13063350—22no—
67C2×#10621884—21no—
68C2×#10621882—21no—
69C2×#10621882—22no—
70C2×#10621882—22no—
71C2×#10621882—22no—
72C2×#10621877C2311yescharacteristic
73C2×#10621877C2311yescharacteristic
74C2×#10621877C2311yescharacteristic
75C2×#10621877C2311yescharacteristic
76C22×C43⋊C8C2311yescharacteristic
77C22×C43⋊C8C2311yescharacteristic
78C4×D172—21no—
79C4×D172—21no—
80C4×D172—21no—
81C4×D172—21no—
82C2×C4⋊Dic43C2311yescharacteristic
83C2×(C2×D43)⋊C4—21no—
84#13063260—22no—
85#13063260—22no—
86#13063258—21no—
87#13063256—21no—
88#13063256—22no—
89C4⋊(C43⋊C8)—22no—
90C4⋊(C43⋊C8)—22no—
91#13063249—21no—
92#13063246—21no—
93#13063246—22no—
94#13063245—21no—
95#13063245—21no—
96#13063245—22no—
97#13063242—22no—
98#13063242—22no—
99#13063241—22no—
100#13063241—22no—
101C22⋊(C43⋊C8)—22no—
102C22⋊(C43⋊C8)—22no—
103#13063236—21no—
104#13063236—21no—
105#13063236D412yes—
106#13063236D412yes—
107C4×C43⋊C8—21no—
108C4×C43⋊C8—21no—
109C2×#41×C43C2311yescharacteristic
110C22×D4×C43C2311yescharacteristic
111C23×C4×C43—21no—
112AGL(1,ℤ/8)×C43—22no—
113AGL(1,ℤ/8)×C43—22no—
114C2×SD16×C43—21no—
115C2×D8×C43—21no—
116C2×M16×C43C2311yescharacteristic
117C2×M16×C43C2311yescharacteristic
118C4×Q8×C43D411yescharacteristic
119C4×Q8×C43D411yescharacteristic
120C4×D4×C43—21no—
121C4×D4×C43—21no—
122C4×D4×C43—21no—
123C4×D4×C43—21no—
124C4×D4×C43—21no—
125C4×D4×C43—21no—
126C4×D4×C43D411yescharacteristic
127C4×D4×C43D411yescharacteristic
128#117×C43—21no—
129#117×C43C2311yescharacteristic
130C2×C4⋊C4×C43C2311yescharacteristic
131C2×C22⋊C4×C43—21no—
132C2×C42×C43—21no—
133C2×C42×C43C2311yescharacteristic
134#107×C43—21no—
135#106×C43—21no—
136#103×C43—21no—
137#102×C43—21no—
138#102×C43—21no—
139#102×C43—21no—
140#97×C43—21no—
141C4×C8×C43—21no—
142C24×C43—21no—
143C23×D43—21no—
144#41×C43—21no—
145#41×C43—21no—
146#41×C43—21no—
147#41×C43—21no—
148C2×Q8×C43C2×D411yescharacteristic
149C2×D4×C43—21no—
150C2×D4×C43—21no—
151C2×D4×C43—21no—
152C2×D4×C43—21no—
153C2×D4×C43—21no—
154C2×D4×C43—21no—
155C2×D4×C43C2×D411yescharacteristic
156C2×D4×C43#4112yes—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43—21no—
161C22×C4×C43—41no—
162C22×C4×C43—41no—
163C22×C4×C43—82no—
164C22×C4×C43#4111yescharacteristic
165C22×C4×C43#4111yescharacteristic
166C22×C4×C43C2411yescharacteristic
167C22×Dic43—21no—
168C2×D172—21no—
169C2×D172—21no—
170C2×D172—21no—
171C2×D172—21no—
172C2×D172—21no—
173C2×D172#4112yes—
174C2×C4×D43—41no—
175C2×C4×D43—41no—
176C2×C4×D43—82no—
177SD16×C43—24no—
178D8×C43—24no—
179M16×C43—21no—
180M16×C43—21no—
181M16×C43—21no—
182M16×C43#4112yes—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43C2×D411yescharacteristic
190C4⋊C4×C43C2×D411yescharacteristic
191C22⋊C4×C43—21no—
192C22⋊C4×C43—21no—
193C22⋊C4×C43—21no—
194C22⋊C4×C43—41no—
195C22⋊C4×C43—41no—
196C42×C43—21no—
197C42×C43—21no—
198C42×C43—21no—
199C42×C43C2×D411yescharacteristic
200C42×C43C2×D411yescharacteristic
201C42×C43C2×D411yescharacteristic
202C42×C43C2×D411yescharacteristic
203#10621884—24no—
204#10621882—28no—
205#10621882—24no—
206(C2×D43)⋊C4—41no—
207(C2×D43)⋊C4—41no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210#10621877—21no—
211#10621877—21no—
212#10621877—21no—
213#10621877—21no—
214#10621877—21no—
215#10621877—21no—
216#10621877—21no—
217#10621877#4112yes—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8C2×D412yes—
227C2×C43⋊C8C2×D412yes—
228C23×C43—21no—
229C23×C43—22no—
230C23×C43—41no—
231C23×C43—41no—
232C23×C43—81no—
233C23×C43—81no—
234C23×C43C2×#4111yescharacteristic
235C22×D43—22no—
236C22×D43—41no—
237C22×D43—41no—
238C22×D43—81no—
239C22×D43—81no—
240Q8×C43—21no—
241Q8×C43—21no—
242D4×C43—22no—
243D4×C43—21no—
244D4×C43—21no—
245D4×C43—22no—
246D4×C43—41no—
247D4×C43—41no—
248D4×C43—41no—
249D4×C43—41no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—81no—
259C2×C4×C43—81no—
260C2×C4×C43—82no—
261C2×C4×C43—82no—
262C2×C4×C43—81no—
263C2×C4×C43#12111yescharacteristic
264C2×C4×C43#12111yescharacteristic
265C2×C4×C43#12111yescharacteristic
266C2×C4×C43#12111yescharacteristic
267C2×C4×C43C22×D411yescharacteristic
268C2×C4×C43C22×D411yescharacteristic
269C2×C4×C43C2×#4111yescharacteristic
270C2×C4×C43C2×#4111yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C2×C4×C43C2×#4111yescharacteristic
273C2×Dic43—41no—
274C2×Dic43—81no—
275D172—22no—
276D172—22no—
277D172—41no—
278D172—41no—
279C4×D43—82no—
280C4×D43—82no—
281C8×C43—22no—
282C8×C43—41no—
283C43⋊C8—22no—
284C43⋊C8—22no—
285C43⋊C8—22no—
286C43⋊C8—41no—
287C43⋊C8—41no—
288C43⋊C8—41no—
289#23719—431nomaximal, Sylow
290C22×C43—21no—
291C22×C43—21no—
292C22×C43—42no—
293C22×C43—41no—
294C22×C43—41no—
295C22×C43—41no—
296C22×C43—82no—
297C22×C43—82no—
298C22×C43—81no—
299C22×C43—81no—
300C22×C43C2×#12111yescharacteristic
301C22×C43#52911yescharacteristic
302C22×C43#52911yescharacteristic
303C2×D43—42no—
304C2×D43—82no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—82no—
308C4×C43—21no—
309C4×C43—21no—
310C4×C43—21no—
311C4×C43—21no—
312C4×C43—41no—
313C4×C43—41no—
314C4×C43—41no—
315C4×C43—81no—
316C4×C43#52511yescharacteristic
317C4×C43#52511yescharacteristic
318C4×C43#52511yescharacteristic
319C4×C43#53211yescharacteristic
320Dic43—81no—
321C4×AGL(1,ℤ/8)—431no—
322C4×AGL(1,ℤ/8)D4311yescharacteristic, complemented
323C2×#442—431no—
324#2923—431no—
325#2090—431no—
326#2079—431no—
327#2063—431no—
328#2063—431no—
329#2051—431no—
330#2051—431no—
331#2045—431no—
332#2018—431no—
333#1778—431no—
334C4×#100—431no—
335C4×#100—431no—
336C2×C43—21no—
337C2×C43—21no—
338C2×C43—81no—
339C2×C43—82no—
340C2×C43#239211yescharacteristic
341C2×C43#306811yescharacteristic
342C2×C43#306811yescharacteristic
343D43—82no—
344C2×AGL(1,ℤ/8)—431no—
345C2×AGL(1,ℤ/8)C2×D4311yescharacteristic
346C22×D8—431no—
347C4×#41C2×D4311yescharacteristic
348C2×C4×D4—431no—
349C2×C4×D4C2×D4311yescharacteristic
350#442—21no—
351#442—862no—
352#442—862no—
353#442—861no—
354#440—21no—
355#440—861no—
356C4×SD16—21no—
357C4×SD16—861no—
358C4×D8—21no—
359C4×D8—861no—
360#433—862no—
361#432—862no—
362C2×#108—431no—
363C2×#108—431no—
364#428—431no—
365#428C2×D4311yescharacteristic
366C2×#106—431no—
367#424—431no—
368#424—431no—
369#418—431no—
370#418C2×D4311yescharacteristic
371#417—431no—
372#417C2×D4311yescharacteristic
373C2×#102—431no—
374C2×#102—431no—
375C2×#100—431no—
376C2×#100—431no—
377C2×#100—431no—
378C2×#100—431no—
379#408—431no—
380#408—431no—
381C4×M16—431no—
382C4×M16—431no—
383C4×M16C2×D4311yescharacteristic
384C2×#97—431no—
385#343—431no—
386#343—431no—
387#343—431no—
388#343—431no—
389#341—431no—
390#341—431no—
391C43#2371911yescharacteristic, complemented, Sylow
392C2×#41C22×D4311yescharacteristic
393C22×D4—431no—
394C22×D4C22×D4311yescharacteristic
395C23×C4—21no—
396C23×C4—861no—
397AGL(1,ℤ/8)—22no—
398AGL(1,ℤ/8)—22no—
399AGL(1,ℤ/8)—862no—
400AGL(1,ℤ/8)—862no—
401C2×SD16—21no—
402C2×SD16—861no—
403C2×D8—21no—
404C2×D8—862no—
405C2×D8—862no—
406C2×D8—862no—
407C2×D8—861no—
408C2×M16—431no—
409C2×M16—431no—
410C2×M16—431no—
411C2×M16—431no—
412C2×M16C22×D4311yescharacteristic
413C2×M16C22×D4311yescharacteristic
414C22×C8—431no—
415C22×C8—431no—
416C4×Q8C22⋊D4311yescharacteristic
417C4×Q8C22⋊D4311yescharacteristic
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—861no—
425C4×D4—861no—
426C4×D4—861no—
427C4×D4—861no—
428C4×D4C22⋊D4311yescharacteristic
429C4×D4C22⋊D4311yescharacteristic
430#117—21no—
431#117C22×D4311yescharacteristic
432C2×C4⋊C4—431no—
433C2×C4⋊C4C22×D4311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—861no—
436C2×C42—21no—
437C2×C42C22×D4311yescharacteristic
438#108—862no—
439#108—862no—
440#107—21no—
441#107—861no—
442#106—21no—
443#106—862no—
444#106—861no—
445C4⋊C8—862no—
446C4⋊C8—862no—
447#103—21no—
448#103—861no—
449#102—21no—
450#102—21no—
451#102—21no—
452#102—862no—
453#102—862no—
454#102—861no—
455#102—861no—
456#102—861no—
457#100—862no—
458#100—862no—
459#100—862no—
460#100—862no—
461C22⋊C8—862no—
462C22⋊C8—862no—
463#97—21no—
464#97—432no—
465#97—432no—
466#97—861no—
467#97—861no—
468C4×C8—21no—
469C4×C8—861no—
470C4×C8—861no—
471C24—21no—
472C24—861no—
473#41—21no—
474#41—21no—
475#41—21no—
476#41—21no—
477C2×Q8C2×C22⋊D4311yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—432no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C2×D4—861no—
489C2×D4—861no—
490C2×D4#1062189812yes—
491C2×D4C2×C22⋊D4311yescharacteristic
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—82no—
499C22×C4—861no—
500C22×C4—3442no—
501C22×C4—1721no—
502C22×C4—1721no—
503C22×C4#1062190011yescharacteristic
504C22×C4#1062190011yescharacteristic
505C22×C4C23×D4311yescharacteristic
506SD16—24no—
507SD16—864no—
508D8—24no—
509D8—868no—
510D8—864no—
511M16—21no—
512M16—21no—
513M16—21no—
514M16—432no—
515M16—861no—
516M16—861no—
517M16—861no—
518M16—861no—
519M16—861no—
520M16—861no—
521M16—861no—
522M16#1062189812yes—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—432no—
526C2×C8—432no—
527C2×C8—861no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C2×C8—861no—
534C2×C8—861no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—861no—
540C4⋊C4—861no—
541C4⋊C4C2×C22⋊D4311yescharacteristic
542C4⋊C4C2×C22⋊D4311yescharacteristic
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—1721no—
549C22⋊C4—1721no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42D4×D4311yescharacteristic
554C42D4×D4311yescharacteristic
555C42C2×C22⋊D4311yescharacteristic
556C42C2×C22⋊D4311yescharacteristic
557C23—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—81no—
562C23—81no—
563C23—862no—
564C23—3441no—
565C23—3441no—
566C23—1721no—
567C23—1721no—
568C23C2×#1062190011yescharacteristic
569Q8—21no—
570Q8—21no—
571D4—21no—
572D4—22no—
573D4—21no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—862no—
580D4—862no—
581D4—1721no—
582D4—1721no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—3442no—
597C2×C4—3442no—
598C2×C4—3441no—
599C2×C4—1721no—
600C2×C4#1306329711yescharacteristic
601C2×C4#1306329711yescharacteristic
602C2×C4#1306329711yescharacteristic
603C2×C4#1306329711yescharacteristic
604C2×C4C2×D4×D4311yescharacteristic
605C2×C4C22×C22⋊D4311yescharacteristic
606C2×C4#41×D4311yescharacteristic
607C2×C4#41×D4311yescharacteristic
608C2×C4C2×#1062189811yescharacteristic
609C2×C4C2×#1062189811yescharacteristic
610C8—22no—
611C8—41no—
612C8—862no—
613C8—862no—
614C8—862no—
615C8—1721no—
616C8—1721no—
617C8—1721no—
618C22—21no—
619C22—21no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—41no—
624C22—82no—
625C22—82no—
626C22—81no—
627C22—81no—
628C22—3442no—
629C22—3442no—
630C22—3441no—
631C22—3441no—
632C22—1722no—
633C22C2×#1306329711yescharacteristic
634C22#1458342911yescharacteristic
635C22#1458343111yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—3441no—
645C4#1458340611yescharacteristic
646C4#1458340811yescharacteristic
647C4#1458341011yescharacteristic
648C4#1458344611yescharacteristic
649C2—21no—
650C2—21no—
651C2—81no—
652C2—82no—
653C2—3442no—
654C2#1913783711yescharacteristic
655C2#1914432011yescharacteristic
656C2#1914432511yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC4×AGL(1,ℤ/8)×C43C2
socleC22×C43C2×#121
prime radical#19135701C2
hypercenterC4×AGL(1,ℤ/8)D43
hypocenterC43#23719
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C4×#13063350C2semidirect
2. C2×#14583009C2semidirect
3. #19143246C2semidirect
4. #19136108C2semidirect
5. #19136043C2semidirect
6. #19135955C2semidirect
7. #19135952C2semidirect
8. #19135885C2semidirect
9. #19135884C2semidirect
10. #19135850C2semidirect
12. #19134502C2semidirect
13. C4×#13063241C2semidirect
14. C4×#13063242C2semidirect
15. C4×AGL(1,ℤ/8)×C43C2semidirect
322. C4×AGL(1,ℤ/8)D43semidirect
391. C43#23719semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#130633502
C2×#145830092
#191432462
#191361082
#191360432
#191359552
#191359522
#191358852
#191358842
#191358502
#191357012
#191345022
C4×#130632412
C4×#130632422
C4×AGL(1,ℤ/8)×C432
#2371943